The Gradient
See Larson 13.6–13.7.
The
derivative $Df$ of a function $f:\mathbb{R}^n\to\mathbb{R}$ (one
output dimension) is called the gradient of $f$, and written
$\vec \nabla f$, when we want to emphasize that the derivative is
a vector field.
You'll want a computer to
help you construct the graphs, particularly $h$. Please use the
Mathematica introduction in Brainhoney. You could use Wolfram
Alpha (use the links in the function chapter if you forgot how
to graph).
Consider the functions $f(x,y)=9-x^2-y^2$, $g(x,y)=2x-y$, and
$h(x,y)=\sin x\cos y$.
-
See
Sage. You can modify these commands to help in the
plots below too.
-
Compute $\vec \nabla g(x,y)$. Then draw both $\vec \nabla g$
and several level curves of $g$ on the same axes.
-
Compute $\vec \nabla h(x,y)$. Then draw both $\vec \nabla h$
and several level curves of $h$ on the same axes.
- What relationships do you see between the gradient vector
field and level curves?
When you present in class, be prepared to provide rough
sketches of the level curves and gradients of each function.
The next problem investigates why the gradient vector and the level curve have the relationship you hopefully saw.
Suppose $\vec r(t)$ is a level curve of $f(x,y)$.
- Suppose you know that at $t=0$, the value of $f$ at $\vec
r(0)$ is 7. What is the value of $f$ at $\vec r(1)$? [What
does it mean to be on a level curve?]
- As you move along the level curve $\vec r$, how much does
$f$ change? Use this to tell the class what
$\ds\frac{df}{dt}$ must equal.
- At points along the level curve $\vec r$, we have the
composite function $f(\vec r(t))$. Compute the derivative
$\ds\frac{df}{dt}$ using the chain rule.
- Use your work from the previous parts to explain why the
gradient always meets the level curve at a 90$^\circ$ angle.
We say that the gradient is normal to level curves
(i.e., a gradient vector is orthogonal to the tangent vector
of the curve).
In the derivative chapter, we extended differential
notation from $dy=f' dx$ to $d\vec y = D\vec f d\vec x$. The key
idea is that a small change in the output variables is
approximated by the product of the derivative and a small change
in the input variables. As a quick refresher, if we have the
function $z=f(x,y)$, then differential notation states that $$dz
= \begin{bmatrix}f_x&f_y\end{bmatrix}
\begin{bmatrix}dx\\dy\end{bmatrix}.$$
Suppose the temperature at a point in the plane is given by the
function $T(x,y)=x^2-xy-y^2$ degrees Fahrenheit. A particle
is at $P=(2,3)$.
- Use differentials to estimate the change in temperature
if the particle moves 1 unit in the direction of $\vec
u=\left(3,4\right)$. [Hint: Find a unit vector in that
direction.]
- What is the actual change in temperature if the particle
moves 1 unit in the direction of $\vec
u=\left(3,4\right)$?
- Use differentials to estimate the change in temperature
if the particle moves about .2 units in the direction of
$\vec u=\left(3,4\right)$.
Let's generalize the calculation we just did into a formula for any $f(x,y)$.
Suppose that $z=f(x,y)$ is a differentiable function (so the
derivative is the matrix
$\begin{bmatrix}f_x&f_y\end{bmatrix}$). Remember to use
differential notation in this problem.
- If $(dx,dy)=(1,0)$, which means we've moved one unit in
the $x$ direction while holding $y$ constant, what is
$dz$?
- If $(dx,dy)=(0,1)$, which means we've moved one unit in
the $y$ direction while holding $x$ constant, what is
$dz$?
- Consider the direction $\vec u=(2,3)$. Find a unit vector
in the direction of $\vec u$. If we move one unit in the
direction of $\vec u$, what is $dz$? [It's all right to leave
you answer as a dot product.]
The directional derivative of $f$ in the direction of the unit
vector $\vec u$ at a point $P$ is defined to be $$D_{\vec u}
f(P) = Df(P) \vec u = \vec \nabla f \cdot \vec u.$$ We dot the
gradient of $f$ with the direction vector $\vec u$. The partial
derivative of $f$ with respect to $x$ is precisely the
directional derivative of $f$ in the $(1,0)$ direction.
Similarly, the partial derivative of $f$ with respect to $y$ is
precisely the directional derivative of $f$ in the $(0,1)$
direction. This definition extends to higher dimensions.
Note that in the definition above, we require the vector
$\vec u$ to be a unit vector. If you are asked to find a
directional derivative in some direction, make sure you start by
finding a unit vector in that direction. We want to deal with
unit vectors because when we say something has a slope of $m$
units, we want to say “The function rises $m$ units if we
run $1$ unit.”
Consider the function $f(x,y) = 9-x^2-y^2$.
- Draw several level curves of $f$.
- At the point $P=(2,1)$, place a dot on your graph. Then
draw a unit vector based at $P$ that points in the direction
$\vec u=(3,4)$ [not to the point $(3,4)$, but in the
direction $\vec u=(3,4)$]. If you were to move in the
direction $(3,4)$, starting from the point $(2,1)$, would the
value of $f$ increase or decrease?
- Find the slope of $f$ at $P=(2,1)$ in the direction $\vec
u=(3,4)$ by finding the directional derivative. This should
agree with your previous answer.
- If you stand at $Q=(-2,3)$ and move in the direction
$\vec v= (1,-1)$, will $f$ increase or decrease? Find the
directional derivative of $f$ in the direction $\vec
v=(1,-1)$ at the point $Q=(-2,3)$.
Recall that the directional derivative of $\vec f$ in the
direction $\vec u$ is the dot product $\vec \nabla f\cdot \vec
u = |\vec \nabla f| |\vec u|\cos\theta$. In this problem,
you'll explain why the gradient points in the direction of
greatest increase.
-
- Why is the directional derivative of $\vec f$ the largest
when $\vec u$ points in the exact same direction as $\vec
\nabla f$? [Hint: What angle maximizes the cosine
function?]
- When $\vec u$ points in the same direction as $\vec
\nabla f$, show that $D_{\vec u}f = |\vec \nabla f|$. In
other words, explain why the length of the gradient is
precisely the slope of $f$ in the direction of greatest
increase (the slope in the steepest direction).
- Which direction points in the direction of greatest
decrease?
Suppose you are looking at a topographical map (see
Wikipedia
for an example). On this topographical map, each contour line
represents 100 ft in elevation. You notice in one section of
the map that the contour lines are really close together, and
they start to form circles around a spot on the graph. You
notice in another section of the map that the contour lines are
spaced quite far apart. Let $f(x,y)$ be the elevation of the
land, so that the topographical map is just a contour plot of
$f$.
For example, you can look at a contour plot of $f(x,y) =
(x+1)^3-3(x+1)^2-y^2+2$ in
Sage.
- At the very top of a mountain, or the very bottom of a
valley, will the gradient be a long vector or a short vector?
Summarizing our results, we have the following theorem.
Let $f$ be a continuously differentiable function, with $\vec
r$ a level curve of the function.
- The gradient is always normal to level curves, meaning
$\vec \nabla f\cdot \dfrac{d\vec r}{dt}=0$.
- The gradient points in the direction of greatest
increase.
- The directional derivative of $f$ in the direction of the
gradient is the length of the gradient. Symbolically, we
write $D_{\vec \nabla f}f = |\vec \nabla f|$.
- At a maximum or minimum, the gradient is the zero
vector.
The next few problems have you practice using
differentials, and then obtain tangent lines and planes to curves
and surfaces using differentials.
The volume of a cylindrical can is $V(r,h)=\pi r^2 h$. Any
manufacturing process has imperfections, and so building a
cylindrical can with designed dimensions $(r,h)$ will result in
a can with dimensions $(r+dr,h+dh)$.
- Compute both $DV$ (the derivative of $V$) and $dV$ (the
differential of $V$).
- If the can is tall and slender ($h$ is big, $r$ is
small), which will cause a larger change in volume: an error
in $r$ or an error in $h$? Use $dV$ to explain your
answer.
- If the can is short and wide (like a tuna can), which
will cause a larger change in volume: an error in $r$ or an
error in $h$? Use $dV$ to explain your answer.
Consider the function $f(x,y)=x^2+y^2$. Consider the level
curve $C$ given by $f(x,y)=25$. Our goal is to find an equation
of the tangent line to $C$ at $P=(3,-4)$.
- Draw $C$. Compute $\vec \nabla f$ and add to your graph
the vector $\vec \nabla f(P)$.
- We know the point $P=(3,-4)$ is on the tangent line. Let
$Q=(x,y)$ represent another point on the tangent line. Add to
your graph the point $Q$ and the vector $\vec {PQ} =
(x-3,y+4)$.
- Why are $\vec \nabla f(P)$ and $\vec{PQ}$ orthogonal? Use
this fact to write an equation of the tangent line.
- What is a normal vector to the line?
This problem might be better if on part 3 we wrote,
“Why does the dot product $\vec \nabla
f(P)\cdot\vec{PQ}$ equal zero? Compute this dot product. Why
is this an equation of the tangent line?” To many
students are correctly giving a vector equation of the
tangent line, by reversing the order of the gradient,
negating one of the terms, and then using (x,y)=dir
vec(t)+point.
The previous problem had you give an equation of the
tangent line to a level curve, by using differential notation.
The next problems asks you to repeat this idea and give an
equation of a tangent plane to a level surface.
Consider the function $f(x,y,z)=x^2+y^2+z^2$. Consider the
level surface $S$ given by $f(x,y,z)=9$. Our goal is to find an
equation of the tangent plane to $S$ at $P=(1,2,-2)$.
- Draw $S$.
- Compute $\vec \nabla f$. Add to your graph the vector
$\vec \nabla f(P)$, with its base at $P$.
- We know the point $P=(1,2,-2)$ is on the tangent plane.
Let $Q=(x,y,z)$ be any other point on the tangent plane. What
is the component form of the vector $\vec {PQ}$?
- Why are $\vec \nabla f(P)$ and $\vec{PQ}$ orthogonal? Use
this fact to write an equation of the tangent plane.
- What is a normal vector to the plane?
Find an equation of the tangent plane to the hyperboloid of one
sheet $1=x^2-y^2+z^2$ at the point $(-3,3,1)$.
The two surfaces $x^2+y^2+z^2=14$ and $3x+4y-z=-1$ intersect in
a curve $C$. Draw both surfaces, and show us the curve $C$.
Then, at the point $(2,-1,3)$, find an equation of the tangent
line to this curve. [Hint: The line is in both tangent planes,
so it is orthogonal to both normal vectors. The cross product
gets you a vector that is orthogonal to two vectors.]
The Second Derivative Test
We start with a review
problems from first-semester calculus.
Let $f(x) = x^3-3x^2$. Find the critical values of $f$ by
solving $f'(x)=0$. Determine if each critical value leads to a
local maximum or local minimum by computing the second
derivative. State the local maxima/minima of $f$. Sketch the
function using the information you discovered.
We now generalize the second derivative test to all
dimensions. We've already seen that the second derivative of a
function such as $z=f(x,y)$ is a square matrix. The second
derivative test relied on understanding if a function was concave
up or concave down. We need a way to examine the concavity of $f$
as we approach a point $(x,y)$ from any of the infinitely many
directions. Such a method exists, and leads to an
eigenvalue/eigenvector problem. I'm assuming that most of you
have never heard the word “eigenvalue.” We could
spend an entire semester just studying eigenvectors. We'd need a
few weeks to discover what they are from a problem-based
approach. Instead, here is an example of how to find eigenvalues
and eigenvectors.
Let $A$ be a square matrix, so in 2D we have $A=\begin{pmatrix}
a&b\\c&d \end{pmatrix}$. The identity matrix $I$ is a
square matrix with 1's on the diagonal and zeros everywhere
else, so in 2D we have $I = \begin{pmatrix} 1&0\\0&1
\end{pmatrix}$. The eigenvalues of $A$ are the solutions
$\lambda$ to the equation $|A-\lambda I|=0$. Remember that
$|A|$ means, “Compute the determinant of $A$.” So
in 2D, we need to find the value $\lambda$ so that
$$\left|\begin{pmatrix} a&b\\c&d\end{pmatrix}-\lambda
\begin{pmatrix} 1&0\\0&1 \end{pmatrix}
\right|=\begin{vmatrix} a-\lambda &b\\c&d-\lambda
\end{vmatrix}=0.$$ This definition extends to any square
matrix. In 3D, the eigenvalues are the solutions to the
equation $$\left|\begin{pmatrix}
a&b&c\\d&e&f\\g&h&i\end{pmatrix}-\lambda
\begin{pmatrix}
1&0&0\\0&1&0\\0&0&1\end{pmatrix}
\right| = \begin{vmatrix}
a-\lambda&b&c\\d&e-\lambda&f\\g&h&i-\lambda\end{vmatrix}=0.$$
An eigenvector of $A$ corresponding to $\lambda$ is a nonzero
vector $\vec x$ such that $A\vec x=\lambda x$.
As you continue taking more upper level science courses (in
physics, engineering, mathematics, chemistry, and more) you'll
soon see that eigenvalues and eigenvectors play a huge role.
You'll start to see them in most of your classes. For now, we'll
use them without proof to apply the second derivative test. In
class, make sure you ask me to show you pictures with each
problem we do, so we can see how eigenvalues and eigenvectors
appear in surfaces.
I would like to put a reference to Ken Kutler's book from
Provo for a proof. I would really like to have more
references period in the book, so if you want to make a
bibtex file, and you ever feel like adding a reference, feel
free.
Let $f(x,y)$ be a function so that all the second partial
derivatives exist and are continuous. The second derivative of
$f$, written $D^2f$ and sometimes called the Hessian of $f$, is
a square matrix.
Because the second derivative is always symmetric (why is
it?), in a linear algebra course you could prove that the
eigenvalues of $D^2f$ must always be real numbers.
Let $\lambda_1$ be the largest eigenvalue of $D^2f$, and
$\lambda_2$ be the smallest eigenvalue. Then $\lambda_1$ is the
largest possible second derivative obtained in any direction.
Similarly, the smallest possible second derivative obtained in
any direction is $\lambda_2$. The eigenvectors give the
directions in which these extreme second derivatives are
obtained. The second derivative test states the following.
Suppose $(a,b)$ is a critical point of $f$, meaning $Df(a,b)
= \begin{bmatrix}0&0\end{bmatrix}$.
- If all the eigenvalues of $D^2f(a,b)$ are positive,
then in every direction the function is concave upwards at
$(a,b)$ which means the function has a local minimum at
$(a,b)$.
- If all the eigenvalues of $D^2f(a,b)$ are negative,
then in every direction the function is concave downwards
at $(a,b)$. This means the function has a local maximum at
$(a,b)$.
- If the smallest eigenvalue of $D^2f(a,b)$ is negative,
and the largest eigenvalue of $D^2f(a,b)$ is positive, then
in one direction the function is concave upwards, and in
another the function is concave downwards. The point
$(a,b)$ is called a saddle point.
- If the largest or smallest eigenvalue of $f$ equals 0,
then the second derivative tests yields no
information.
Consider the function $f(x,y)=x^2-2x+xy+y^2$. The first and
second derivatives are
$$Df(x,y)=\begin{bmatrix}2x-2+y,x+2y\end{bmatrix}
\quad\text{and}\quad D^2f = \begin{bmatrix}2&1
\\1&2\end{bmatrix}.$$ The first derivative is zero (the
zero matrix) when both $2x-2+y=0$ and $x+2y=0$. We need to
solve the system of equations $2x+y=2$ and $x+2y=0$. Double the
second equation, and then subtract it from the first to obtain
$0x-3y=2$, or $y=-2/3$. The second equation says that $x=-2y$,
or that $x=4/3$. So the only critical point is $(4/3,-2/3)$.
In this example, the second derivative is constant, so the
point $(4/3,-2/3)$ did not change the matrix. In general, the
point will affect your matrix. See
Sage
to see a graph which shows the eigenvectors in which
the largest and smallest second derivatives occur.
We find the eigenvalues of $D^2 f(4/3,-2/3)$ by solving
the equation $$\begin{vmatrix}2-\lambda&1
\\1&2-\lambda\end{vmatrix} = (2-\lambda)(2-\lambda)-1=0.$$
Expanding the left hand side gives us {$4-4\lambda + \lambda^2
-1 = 0$}. Simplifying and factoring gives us
$\lambda^2-4\lambda +3 = (\lambda-3)(\lambda -1) = 0$. This
means the eigenvalues are $\lambda = 1$ and $\lambda=3$. Since
both numbers are positive, the function is concave upwards in
every direction. The critical point $(4/3,-2/3)$ corresponds to
a local minimum of the function. The local minimum is the
output $f(4/3,-2/3) = (4/3)^2-2(4/3)+(4/3)(-2/3)+(-2/3)^2$. A
graph of $f$ is provided on the right. The red vector $(1,1)$
points in the direction in which the second derivative is the
largest value 3. The red vector $(-1,1)$ points in the
direction in which the second derivative is the smallest value
1. These vectors are called eigenvectors, and you can learn
much more about them, in particular how to find them, in a
linear algebra course. For this course, we just need to be able
to find eigenvalues.
See 14.7 for more practice.
Consider the
function $f(x,y)=x^2+4xy+y^2$.
- Find the critical points of $f$ by finding when $Df(x,y)$
is the zero matrix.
- Find the eigenvalues of $D^2f$ at any critical
points.
- Label each critical point as a local maximum, local
minimum, or saddle point, and state the value of $f$ at the
critical point.
Consider the function $f(x,y)=x^3-3x+y^2-4y$.
- Find the critical points of $f$ by finding when $Df(x,y)$
is the zero matrix.
- Find the eigenvalues of $D^2f$ at any critical points.
[Hint: First compute $D^2f$. Since there are two critical
points, evaluate the second derivative at each point to
obtain 2 different matrices. Then find the eigenvalues of
each matrix.]
- Label each critical point as a local maximum, local
minimum, or saddle point, and state the value of $f$ at the
critical point.
Consider the function $f(x,y)=x^3 + 3xy +y^3$.
- Find the critical points of $f$ by finding when $Df(x,y)$
is the zero matrix.
- Find the eigenvalues of $D^2f$ at any critical
points.
- Label each critical point as a local maximum, local
minimum, or saddle point, and state the value of $f$ at the
critical point.
\label{optimize box in cake} For my daughter's birthday, she
has asked for a Barbie princess cake. I purchased a metal pan
that's roughly in the shape of a paraboloid
$z=f(x,y)=9-x^2-y^2$ for $z\geq 0$. To surprise her, I want to
hide a present inside the cake. The present is a bunch of small
candy that can pretty much fill a box of any size. I'd like to
know how large (biggest volume) of a rectangular box I can fit
under the cake, so that when we start cutting the cake, she'll
find her surprise present. The box will start at $z=0$ and the
corners of the box (located at $(x,\pm y)$ and $(-x,\pm y)$)
will touch the surface of the cake $z=9-x^2-y^2$.
- What is the function $V(x,y)$ that we are trying to
maximize?
- If you find all the critical points of $V$, you'll
discover there are 9. However, only one of these critical
points makes sense in the context of this problem. Find that
critical point.
- Use the second derivative test to prove that the critical
point yields a maximum volume.
- What are the dimensions of the box? What's the volume of
the box?
The only thing left for me is to now determine how much
candy I should buy to fill the box. I'll take care of that.
When was in Provo, I remember teaching a business calc course.
In the text, I recall reading about an aerospace company (maybe
Boeing) that tried to build an “optimal” wing. They
set up their equations, found derivatives, and build the wing
that the critical point suggested they should build. Turns out
they build the “worst” wing instead of the
“best.” This was just a classic case of why you
should ALWAYS check if the critical points lead to maxes or
mins. I can't put this in the text until I find the reference.
I want to add it to the text someday.
In this problem, we'll derive the version of the second
derivative test that is found in most multivariate calculus
texts. The test given below only works for functions of the form
$f:\mathbb{R}^2\to\mathbb{R}$. The eigenvalue test you have been
practicing will work with a function of the form
$f:\mathbb{R}^n\to\mathbb{R}$, for any natural number $n$.
Suppose that $f(x,y)$ has a critical point at $(a,b)$.
- Find a general formula for the eigenvalues of
$D^2f(a,b)$. Your answer will be in terms of the second
partials of $f$.
- Let $D=f_{xx}f_{yy}-f_{xy}^2$.
- If $D<0$, explain why $f$ has a saddle point at
$(a,b)$.
- If $D=0$, explain why the second derivative test
fails.
- If $D>0$, explain why $f$ has either a maximum or
minimum at $(a,b)$.
- If $D>0$, and $f_x(a,b)>0$, does $f$ have a
local max or local min at $(a,b)$. Explain.
- The only critical point of $f(x,y) = x^2+3xy+2y^2$ is at
$(0,0)$. Does this point correspond to a local maximum, local
minimum, or saddle point? Give the eigenvalues (which should
come instantly out of part 1). Find $D$, from part 2, to
answer the question.
Lagrange Multipliers
The last problem was an example of
an optimization problem where we wish to optimize a function (the
volume of a box) subject to a constraint (the box has to fit
inside a cake). If you are economics student, this section may be
the key reason why you were asked to take multivariate calculus.
In the business world, we often want to optimize something
(profit, revenue, cost, utility, etc.) subject to some constraint
(a limited budget, a demand curve, warehouse space, employee
hours, etc.). An aerospace engineer will build the best wing that
can withstand given forces. Everywhere in the engineering world,
we often seek to create the “best” thing possible,
subject to some outside constraints. Lagrange discovered an
extremely useful method for answering this question, and today we
call it “Lagrange Multipliers.” Rather than introduce
Cobb-Douglass production functions (from economics) or
sheer-stress calculations (from engineering), we'll work with
simple examples that illustrate the key points. Sometimes silly
examples carry the message across just as well.
Suppose an ant walks around the circle $g(x,y)=x^2+y^2=1$. As
the ant walks around the circle, the temperature is $f(x,y) =
x^2+y+4$. Our goal is to find the maximum and minimum
temperatures reached by the ant as it walks around the circle.
We want to optimize $f(x,y)$ subject to the constraint
$g(x,y)=1$.
- Draw the circle $g(x,y)=1$. Then, on the same set of
axes, draw several level curves of $f$. The level curves
$f=3, 4, 5, 6$ are a good start. Then add more (maybe at each
1/4th). If you make a careful, accurate graph, it will help a
lot below.
- Based solely on your graph, where does the minimum
temperature occur? What is the minimum temperature?
- If the ant is at the point $(0,1)$, and it moves left,
will the temperature rise or fall? What if the ant moves
right?
- On your graph, place a dot(s) where you believe the ant
reaches a maximum temperature (it may occur at more than one
spot). Explain why you believe this is the spot where the
maximum temperature occurs. What about the level curves tells
you that these spots should be a maximum.
- Draw the gradient of $f$ at the places where the minimum
and maximum temperatures occur. Also draw the gradient of $g$
at these spots. How are the gradients of $f$ and $g$ related
at these spots?
Suppose $f$ and $g$ are continuously differentiable functions.
Suppose that we want to find the maximum and minimum values of
$f$ subject to the constraint $g(x,y)=c$ (where $c$ is some
constant). Then if a maximum or minimum occurs, it must occur
at a spot where the gradient of $f$ and the gradient of $g$
point in the same, or opposite, directions. So the gradient of
$g$ must be a multiple of the gradient of $f$. To find the
maximum and minimum values (if they exist), we just solve the
system of equations that result from $$\vec \nabla f = \lambda
\vec \nabla g,\quad \text{and}\quad g(x,y)=c$$ where $\lambda$
is the proportionality constant. The maximum and minimum values
will be among the solutions of this system of equations.
Suppose an ant walks around the circle $x^2+y^2=1$. As the ant
walks around the circle, the temperature is $T(x,y) = x^2+y+4$.
Our goal is to find the maximum and minimum temperatures $T$
reached by the ant as it walks around the circle.
- What function $f(x,y)$ do we wish to optimize? What is
the constraint $g(x,y)=c$?
-
The most common error on this problem is to divide both
sides of an equation by $x$, which could be zero. If you
do this, you'll only get 2 ordered pairs.
% Find the gradient of $f$ and the gradient of $g$.
Then solve the system of equations that you get from the
equations $$\vec \nabla f = \lambda \vec \nabla g, \quad
\quad x^2+y^2=1.$$ You should obtain 4 ordered pairs
$(x,y)$.
- At each ordered pair, find the temperature. What is the
maximum temperature obtained? What is the minimum temperature
obtained.
\marginparbmw{See 14.8 for more practice.}
Consider the curve
$xy^2=54$ (draw it). The distance from each point on this curve
to the origin is a function that must have a minimum value.
Find a point $(a,b)$ on the curve that is closest to the
origin. [The distance to the origin is $d(x,y)=\sqrt{x^2+y^2}.$
This distance is minimized when $f(x,y) = x^2+y^2$ is
minimized. So use $f(x,y)=x^2+y^2$ as the function you wish to
minimize. What's the constraint $g(x,y)=c$?]
Find the dimensions of the rectangular box with maximum volume
that can be inscribed inside the ellipsoid
$$\frac{x^2}{2^2}+\frac{y^2}{3^2}+\frac{z^2}{5^2}=1.$$ [What is
the function $f$ you wish to optimize? What is the constraint
$g=c$? Try solving each equation for $\lambda$ so you can
eliminate it from the problem.]
Repeat problem
TODO,
but this time use Lagrange multipliers. Find the dimensions of
the rectangular box of maximum volume that fits underneath the
surface $z=f(x,y)=9-x^2-y^2$ for $z\geq 0$. [Hint: Let
$f(x,y,z) = (2x)(2x)(z)$ and $g(x,y,z)=z+x^2+y^2=9$. You'll get
a system of 4 equations with 4 unknowns ($x,y,z,\lambda$). Try
solving each equation for lambda. You know $x,y,z$ can't be
zero or negative, so ignore those possible cases.]
This would be an appropriate place to put Taylor series in high
dimensions, if your school wants the students to see those
ideas. You can introduce it all using matrix derivatives. You
could then actually prove the second derivative test, as well
as define positive definite matrices.
We finished all the problems in this unit in 4 days of time
(really only 3 and a half, as the line integral unit overlapped
on the first day.